Dynamics of a reaction-diffusion SIRS model with general incidence rate in a heterogeneous environment

Abstract

In this paper, we study a diffusive SIRS-type epidemic model with transfer from the infectious to the susceptible class. Our model includes a general nonlinear incidence rate and spatially heterogeneous diffusion coefficients. We compute the basic reproduction number $R_0$ of our model and establish the global stability of the disease-free steady state when $R_0<1$ . Furthermore, we study the uniform persistence when $R_0>1$ and perform a bifurcation analysis for a special case of our model. Some numerical simulations are presented to illustrate the dynamics of the solutions as the model parameters are varied.

Keywords: Bifurcation analysis; Global stability; Reaction–diffusion; Uniform persistence.